Rational Indices
Understand rational indices for Queensland Year 11 Mathematical Methods (QCAA). A fractional power is just another way of writing a root: a power of one over n means the nth root, linking index notation to surds.
You will learn to read a fractional index as a root, evaluate these powers, handle negative rational indices, and convert between surd form and index form — extending the index laws to every rational power.
Every question with a fully worked solution.
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Theory
In Year 11 Mathematical Methods (QCAA, Unit 2), a rational (fractional) index means a root: \(a^{1/n}\) is the \(n\)th root of \(a\), and \(a^{m/n}\) raises that root to the power \(m\). This page shows how to evaluate fractional and negative rational indices and how to convert between surd (radical) and index form.
A rational index is an index that is a fraction. The denominator gives a root and the numerator gives a power: \(a^{1/n}=\sqrt[n]{a}\) is the number that, raised to the power \(n\), gives \(a\).
More generally \(a^{m/n}=\left(\sqrt[n]{a}\right)^{m}=\sqrt[n]{a^{m}}\). Taking the root first usually keeps the numbers small. A negative rational index combines this with the reciprocal rule, \(a^{-m/n}=\dfrac{1}{a^{m/n}}\). All the ordinary index laws still apply.
For a positive base \(a\) and positive integers \(m,\,n\):
How to evaluate \(a^{m/n}\)
- Root first: take the \(n\)th root of the base (the denominator tells you which root), so \(\sqrt[n]{a}\).
- Then the power: raise that result to the power \(m\) (the numerator).
- Handle a negative index last: if the index is negative, take the reciprocal at the end, \(a^{-m/n}=\dfrac{1}{a^{m/n}}\).
A \(\tfrac13\) index means the cube root:
| \(64^{1/3}\) | \(=\) | \(\sqrt[3]{64}\) |
Find the number that cubes to \(64\):
| \(=\) | \(4\quad(\text{since }4^{3}=64)\) |
\(64^{1/3}=4\).
Denominator \(3\) is the root, numerator \(2\) is the power — root first:
| \(27^{2/3}\) | \(=\) | \(\left(\sqrt[3]{27}\right)^{2}\) |
| \(=\) | \((3)^{2}\) |
Now apply the power:
| \(=\) | \(9\) |
\(27^{2/3}=9\).
Negative index — take the reciprocal:
| \(16^{-3/4}\) | \(=\) | \(\dfrac{1}{16^{3/4}}\) |
Evaluate \(16^{3/4}\): fourth root, then cube:
| \(16^{3/4}\) | \(=\) | \(\left(\sqrt[4]{16}\right)^{3}\) |
| \(=\) | \((2)^{3}=8\) |
Substitute back:
| \(=\) | \(\dfrac{1}{8}\) |
\(16^{-3/4}=\dfrac{1}{8}\).
Convert each surd to index form:
| \(\sqrt[3]{x^{2}}\) | \(=\) | \(x^{2/3}\) |
| \(\sqrt[3]{x}\) | \(=\) | \(x^{1/3}\) |
Product law — add the indices:
| \(x^{2/3}\times x^{1/3}\) | \(=\) | \(x^{\tfrac{2}{3}+\tfrac{1}{3}}\) |
| \(=\) | \(x^{1}\) | |
| \(=\) | \(x\) |
\(\sqrt[3]{x^{2}}\times \sqrt[3]{x}=x\).
Common pitfalls
Frequently asked questions
What does a fractional index mean?
A fractional index is a root: \(a^{1/n}=\sqrt[n]{a}\). More generally \(a^{m/n}=\left(\sqrt[n]{a}\right)^{m}\), where the denominator is the root and the numerator is the power.
How do you evaluate a^(m/n)?
Take the \(n\)th root of the base first, then raise the result to the power \(m\). For \(27^{2/3}\), \(\sqrt[3]{27}=3\) then \(3^{2}=9\).
What does a negative fractional index give?
Take the reciprocal of the positive-index value: \(a^{-m/n}=\dfrac{1}{a^{m/n}}\). For example \(16^{-3/4}=\dfrac{1}{8}\).
How do you convert a surd to index form?
Write the root as the denominator of the index: \(\sqrt{x}=x^{1/2}\) and \(\sqrt[n]{x^{m}}=x^{m/n}\).
Why take the root before the power?
Rooting first keeps the numbers small, so \(27^{2/3}=(\sqrt[3]{27})^{2}=3^{2}=9\) is easier than cubing \(27\) first.